math

=LESSONS LEARNED THROUGH IMPLEMENTATION= = =

=="Mathematics is used to describe and explain relationships. As part of the study of mathematics, students look for relationships among numbers, sets, shapes, objects and concepts. The search for possible relationships involves collecting and analyzing data and describing relationships visually, symbolically, orally or in written form." ==

//Program of Studies: (K to 9 document, p 12, 10 to 12 document, p.) //
==My response as a teacher then is to make evident to students that our goal is to find and explain relationships because that is how you LEARN about YOURSELF as a LEARNER. ==

The problem was stated as Lisa is making bracelets for her 12 friends. She has 8 finished, how many more to make?
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= = = = =These students all have a piece of knowledge about 8 + 4.= ==But how do they translate that "known" fact into the solution of this problem. The problem as posed suggests subtraction might be involved. Do we need students to arrive at a subtraction equation? == = = = = = = = = = = = = = = = =

=LESSON LEARNED: = ===Teaching and learning are complex, highly dynamic processes. We must move away from linear models for covering curriculum in isolated subjects… Below is an organizer developed to help teachers balance the key ideas that drive their planning and instruction. We must honour the research on effectiveness, we must consider inclusion and differentiation, we must link curriculum to competencies. And we must teach teachers to engage in PD that is focused on learning, not covering programs, not getting activities but LEARNING and LEARNING TO LEARN. === ==

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//The cohort groups experimented with a number of models for infusing the competencies into their thinking about planning. We started with the Framework for Student Learning document, available on line from Alberta Education. The document below is our latest reworking to mesh with the language used in the Ministerial Order on Student Learning, May 13, 2013.//======

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=A TEACHER QUESTION THAT IMMEDIATELY EMERGES: =

===// Where do we start if we are going to have students learn to take control of, responsibility for the assessment of and the motivation to continually improve their abilities to learn? Our math curriculum clearly states: //===

// "To experience success, students must be taught to set achievable goals and assess themselves as they work toward these goals. // // Striving toward success and becoming autonomous and responsible learners are ongoing, reflective processes that involve revisiting the ////setting and assessing of personal goals." //

START WITH ATTITUDES AND DISPOSITIONS
// "Students with positive attitudes toward learning mathematics are likely to be motivated and prepared to learn, participate willingly in classroom // // activities, //// persist in challenging situations and engage in reflective practices." //

===// TEACHERS deliberately engage students in experiences and discussions that unpack what it means to be a learner, what it means to learn and descriptions of the attitudes and dispositions of a lifelong learner. The attached sheet is intended to simply promote a conversation around dispositions… Spend time with your students building, adapting and refining statements that they can use to describe themselves and their behaviours in each category. //===



==Teachers use student samples to fuel discussions around what it looks like when someone is communicating clearly, describing or explaining their thinking so that we can understand it. Here is a student sample, that might be used. ==

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=WHEN TEACHERS CHANGE THEIR PRACTICE THEY CHANGE THEIR PLANNING DOCUMENTS =

====//Teacher planning documents become dynamic and fluid. It is critical that teachers "know" curriculums in a rich and connected way. Weaving together process and content outcomes, infusing front matter and goal statements into their thinking. Pd sessions that include direct and constant attention to specific curriculum statements, that push teachers to make direct references to statements in the document, connected to planning templates that include key ideas that should direct planning have helped teachers begin the process. BUT LINEAR DOCUMENTS AND SYSTEMS FOR RECORDING ARE STANDING IN THE WAY OF TEACHER PROGRESS.//====

Samples that have emerged over the past 6 years:







= = = = = = =Mathematical Reasoning is built on number properties and logical thinking: =

===Equality: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Introduced as early as Kindergarten by the statement match my set. Clearly stated in Grades 1 and 2. Understanding equal in terms of both sides of the equation describe the same quantity and therefore can be considered "balanced ===

===<span style="color: #ff00ff; font-family: 'Comic Sans MS',cursive; font-size: 130%;">If Then logic: <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Knowing facts is only of value if I can apply them to strategize and solve problems. Memorizing all the knowledge in the world is silly. Knowing a few key "facts" and strategies for quickly manipulating them is the goal. Then lots of practice with puzzling and using them will keep them automatic. ===

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===<span style="color: #ff00ff; font-family: 'Comic Sans MS',cursive; font-size: 130%;">Multiplication //<span style="font-family: 'Times New Roman',Times,serif;"> i //<span style="font-family: 'Times New Roman',Times,serif;">s not JUST REPEATED ADDITION. A lack of multiplicative reasoning is stopping students from progressing in the upper grades. WE MUST INFUSE OUR TEACHING WITH MODELS THAT BRING ATTENTION TO THE LINKS BETWEEN FRACTIONS, DECIMALS, PLACE VALUE, DIVISION AND MULTIPLICATION. Multiplication grows dimensionally making transformations an important idea. Moving things in your mind and considering how the move affected the quantity and the properties of the item. ===

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=<span style="font-family: 'Comic Sans MS',cursive; font-size: 130%;">LESSON LEARNED: Influencing the Field = ===//<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Pre Service Teachers are an influential audience and potential conduit for infusing Curricular Change and the Competencies into the field. I have embedded attention to the Competencies, Inspiring Education and the Student Framework documents into my work with EDEL math and science students. They embrace the intent, make connections and actively search them out in their field experiences, in their practicums and in their discussions with members of the Public. Below are excerpts from the writings of several students to emphasize this point. //===

Sample 1
 * //<span style="color: #b51010; font-family: 'Times New Roman',Times,serif;">Today I participated in a personal development day with James Tanton. I was afraid to go because it was working with polynomials, which I haven’t done in awhile, but I am glad I did. James clearly made the front matter of the program of studies come to life. What struck me immediately was how clearly his teaching modeled that “Learning through problem solving (can) be the focus of mathematics at all grade levels. When students encounter new situations and respond to questions of the type, How would you…? Or How could you… the problem-solving approach is being modeled. Students develop their own problem-solving strategies by listening to, discussing and trying different strategies.” (Alberta Program of Studies Math, Pg. 8) Teaching math in ways that meet the learning needs of every student is a challenge but I could certainly connect to James’ modeling. His confidence made me realize that as a math teacher I have to exude confidence as well. //

//<span style="color: #b51010; font-family: 'Times New Roman',Times,serif;"> James expected us to reason and think things out, he encouraged us to follow logic when answering and used prompting and guided questions to direct us. This certainly links to the program of studies; “Mathematical reasoning helps students think logically and make sense of mathematics. Students need to develop confidence in their abilities to reason and justify their mathematical thinking.” (POS, Pg. 8) By letting us feel that what we had to say was worth listening to, clarifying and discussing, he modeled how teachers might help students learn toconsider the context and seek additional information and perspectives when analyzing information. (Making them) able to reflect on their learning, recognizing strengths and weaknesses in their reasoning and in arguments presented by others.(Inspiring Education, p10) //

//<span style="color: #b51010; font-family: 'Times New Roman',Times,serif;">A connection I made with the curriculum, start from where kids are at. James would always start from a place we could all relate to and then gently push us deeper and deeper into the concept. He honoured and accepted what we knew and if after probing it appeared we were confident, then he would challenge us to go further, try something harder or find the next level in the sequence. I think this links to helping students become critical thinkers: // //<span style="color: #b51010; font-family: 'Times New Roman',Times,serif;">“As critical thinkers, (students) use metacognition to reflect on their thinking and recognize strengths and weaknesses in their reasoning and in the positions presented by others. Student have the confidence and capacity to solve a range of problems, from simple to complex and including novel to ill-defined, related to their learning, their work or their personal lives.” (Framework for Student Learning, Pg. 3) // //<span style="color: #b51010; font-family: 'Times New Roman',Times,serif;">They can identify and predict problems and solutions that are not readily apparent. These students are aware of and can use multiple approaches to solving a problem, including collaboration.( Inspiring Education, p10) // ||

The entire transcripts is in this document.

Sample 2
 * //<span style="font-family: 'Times New Roman',Times,serif; font-size: 130%;">His presentations combine specific and focused content from curriculum with a delivery that exemplifies a problem solving approach. Specifically evident, his reliance on visual, spatial models, the search for and explanation of patterns in data and the care he takes to link the models and visuals to the symbolic. James included several in class demonstrations in how workshops, allowing teachers to listen and watch as students engaged with the material. A comment repeated by many participants: "I was and am amazed at how much content you covered in such a short space of time… I see how students would be engaged and eager to participate. But more amazing, I have been spending way too long on isolated problems and activities in the textbook when you were able to so quickly get them to the real point of the concept." // ||

Sample 3
 * //<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">When I got home I decided to teach my husband what I learned. Math was his best subject in university and he understood polynomials without feeling the need to make sense of arrays but he wanted to know how math teaching had changed so he persevered. As we were working, he asked how come the math curriculum changed anyway. He used the argument I have heard often: “Why change something that wasn’t broken. What’s wrong with the way we have been doing math?” //

//<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">I changed his mind with one question, “Was math difficult for you in school?” //

//<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">He said he always loved math, and then I proceeded to tell him how much trouble I had with math in school. I could never do it the way I understood and that was not allowed. I had to learn it the way the teacher said. Today students are allowed to use their own thinking and to explain how they think and how they get their answers. I shared the idea of B.E.R.C.S. with him as a way for children to build their understanding by working from they build, explain and represent. It allows them to use their own words and come to believe in themselves. The compare is where they see other ways and begin to make sense of whether or not what they are thinking is “enough” or makes sense. The more they compare and discuss and engage with others in how they are doing the same math, the more they practice and develop mental ideas and strategies. They “know” what they have learned. //

//<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">My husband saw the value in allowing every child to be successful with math not just the ones like him who seemed to just get it. He was beginning to understand that differentiation is key in today’s society. //

//<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">It was when I showed him how to use the N1 and N2 that we really made a difference even to his understanding. Amazing as it sounds, He actually said to me and I quote: “It this math was taught to me in high school, it would have made learning easier.” // ||

= = =<span style="font-family: 'Comic Sans MS',cursive; font-size: 120%;">Teacher who are deliberately working on change are : = =<span style="font-family: 'Comic Sans MS',cursive; font-size: 120%;"> INCLUDING STUDENTS IN THE CHANGE PROCESS = = = // As teachers and student teachers studied the Inspiring Education document, and the Framework for Student Learning they began to search out ways to help their students "Know How to Learn " and models that they could use to provide their students with language to describe and evaluate their own behaviours, attitudes and actions in coming to "know themselves as learners" as well as learning to set goals for improving their learning. //.

===STUDENTS PARTICIPATED IN PD opportunities with teachers. Throughout the year, on several occasions junior high students and Elementary Education Pre SErvice teachers participated in professional development workshops with teachers. Their questions and insights provided a unique perspective for teachers to consider as they learned. Cohort teachers regularly share what they are learning with their classes and several report their students look forward to what they might bring back from meetings.===

From PD with PWSD

We discussed __**Risktaking.**__ What do we want students to see, believe, understand about risk taking. What do we as teachers need to model and encourage if our students are to become risk takers? =How do we encourage risk taking? By accepting every response and gently discouraging the "sillies". When a student says who cares or what does it matter I turn the question back. What does it matter? Who might care? When a silly answer comes, we laugh and I tell students that first answers often just break the tension or get everyone more relaxed with a laugh. Now as you really start thinking, let's hear your thoughts.=

=WE MUST HONOUR AND VALUE WAIT TIME AND WE MUST TELL STUDENTS THAT IS WHAT WE ARE DOING. Do not raise your hand right away. I am going to give everyone time to think…. everyone needs to be sure they heard the question and really thought about it. If you are afraid you will forget, write it down while you wait. CURRENT RESEARCH SUGGESTS THAT SOME STUDENTS NEED AS MUCH AS 2 full minutes TO PROCESS THE QUESTION AND organize their response. Tell students thinking takes a few minutes. Do not blurt out your first response, wait and think…=

=We discussed the fear and anxiety students feel in raising their hands and giving answers which begs the question why? What have their experiences been to this point that they are afraid to join in a learning community? How can we change that?=

=TEACHERS MUST MODEL thinking with talk alouds, making and correcting mistakes in front of students, talking about your own mistakes in thinking or sharing and comparing what you heard and wondered about…" //One of the other teachers said or at a workshop I was told… what do you think class? Has this every happened to you???"//=

==//<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Here is a sample of a student's thinking to solve a problem. Unfortunately I suspect the bats created a diversion and more time went into drawing bats than thinking about 8. This is a problem we will have with trying to create word problems that young students can actually use to access important mathematical ideas. My goal with this problem in Grade one is to find out: Does the student have imagery for 8 that he or she can draw on to "know" a starting point for how to break 8 into two pieces. So when I say 8 do you think about the parts that make 8 and what parts do you think about. I would expect a Grade one to have 4 plus 4 as an automatic. I would also expect some might have 7 plus 1 but not all would. If we had done lots of dot collection discussions I would also expect to hear 5 plus 3 or 6 plus 2. I hope this information would be available in memory as soon as the student understood the task. Unfortunately, Grade ones often are sidetracked by the drawing being accurate or the concentration they need to do any work on paper and so we seldom see the true evidence of all they know. In this case, I have no idea why he chose 2 and 6. But I suspect he started drawing bats, the time it took was so much effort he moved to balls. What I want to know is did he think 2 and 6 will work, then start drawing? //==



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//The problem reads I have 8 balls and bats. How many of each could I have. Keep track of your work with pictures, number and or words. I might change the instruction to use diagrams, words, numbers to explain where you started and how you solved… The instructions are very dependent on what prior work you have done with students…. I would never just hand this sheet out… I have to use instructions my students are familiar with and this is often why when we find "worksheets" from internet sources and simply reproduce them we do not get the result we had hoped for. WE must build comprehension with students, teaching them how to read for information and recognize information in text outside of actually problem solving.//=====

=I would expect in Grade 2 that students could start explaining to me how many different solutions there might be to this question.= =If you have 8 things in the bag and some are bats and some are balls how many of each might there be?= =How will you know when you have found all the possibilities?= ====//Grade 2 students are expected to sort by 2 attributes… to be able to keep 2 separate ideas in play. So make 8 with 2 addends, how many different ways. Bats and balls so 2 bats and 6 balls is different from 2 balls and 6 bats….//====

=Grade 2 and 3 need to discuss and compare solutions to see that there is a list that will emerge…= ====//If the list comes automatically, this is no longer a problem and the student needs to be moved to a real problem. For example a student starts at 1 and 7, and runs down the list in a counting sequence 2 and 6, 3 and 5, 4 and 4, 5 and 3, 6 and 2, 7 and 1 because he or she knows that is how you find all the addends is likely not problem solving. This is not a problem, it is a procedure.//==== ====//Simply by asking: "Will that always work, for any number?" a teacher might force a student to new levels of thinking. If the student is firmly convinced, yes, it always works it is time for a different challenge.//====

=As student generate solutions, posting the data on the board is a place to begin a discussion of how we need to organized data to search for relationships...=

=<span style="color: #ff00ff; font-family: 'Comic Sans MS',cursive; font-size: 130%;">BUILD EXPLAIN REPRESENT COMPARE =

=__**Organizing data**__ is a critical thinking skill and one we should be encouraging and designing lessons to develop. On Friday you were given a bag of blocks already organized into chunks of 2 and chunks of 3. You were asked not to unsnap the chunks. The problem posed was to snap together 17, quick as you can while alternating colours.=



=//As people finished the task, I asked you to explain what you thought about when you started: Some said I thought 5 and 5 and 5 is 15 so I snapped together 2s and 3s thinking 5. Others thought 2 and 3 is 5, 2 is 7, 3 is 10, 2 and 3 is 15 and 2 is 17. Others thought I need 10 and 7 and I can make 10 with two fives so 2 and 3 and 2 and 3. Now i need 5 and 2 so 2 and 3 and 2.//=

=//Some just started snapping and then realized they needed to count to know if they had 17.//=

=//STUDENTS MUST BE CONSTANTLY AND CONSISTENTLY REMINDED our goal in math is to minimize the count. Your brain knows 2 and 3 without counting. That is why I gave you 2s and 3s. With a little practice (we have been doing this since Kindergarten) your brain knows 2 and 3 is five. DO NOT COUNT EVERY BLOCK when you have chunks you can trust. So any student who goes back and counts every block needs to be directed to thinking in chunks.//=

=As students explained their thinking I started putting expressions on the board. So first person said they thought:=

=5 + 5 + 5 + 2=

=This is a way to EXPRESS or describe the number 17. I titled my list 17. We are building expressions for 17.=

=(//The word expression is in the curriculum as early as Grade 2. We need to use it to describe numbers)//=

=//Here is a picture of the list we developed. I used brackets as we described Chunks that match. So I see 9 and 8 because I can see 2 + 2 + 2 + 2 is 8 and 3 + 3 + 3 is 9. I MEAN I CAN SEE IT IN THE BLOCKS.//=

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=//LINKING THE ACTUAL BLOCKS TO THE EXPRESSIONS IS A CRITICAL PIECE. We must allow students to make the link between what they see and what we write. I encourage moving in your head, but often a student needs to come up and literally move the blocks around to show the class what they saw in their heads.//=

=//Visualization must be prompted, encouraged and valued !!!!!!//=

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=//Can you see each of my expressions in this visual representation of 17. You do not need to draw it and in fact I believe if we ask students to draw this right now we lose the value of the exercise altogether. I want you to clearly see 2 and 3 is 5 and 5 and 5 is ten in this image…. I see://=

=//2 + 3 + 2 + 3 + 2 + 3 + 2//=

=//5 + 5 + 5 + 2//=

=//2 + 2 + 2 + 2 + 3 + 3 + 3//=

=//8 + 9//=

=//10 + 7//=

=//Now I see 12 + 5 cause I put 10 and 2 together and that left a 5.//=

=//Which of these is related (they all are but some seem to come together automatically.)//=

=//(2 + 3 )+ (2 + 3) + (2 + 3) + 2//=

=//5 + 5 + 5 + 2//=

=//then I saw//=

=//(5 + 5) + ( 5 + 2)//= //10 + 7//=

=//then I saw//=

=//(5 + 5 + 5) + 2//= //15 + 2//=

=**WHAT IS VITALLY IMPORTANT HERE IS THAT EVERY STUDENT CAN LITERALLY SEE EACH OF THESE COMBINATIONS IN THE BLOCKS. THIS IS NOT TRY TO REMEMBER OR COUNTING WORK. SEE THE GROUPS IN THE BLOCKS AND IF YOU ARE STRUGGLING TO SEE, RE ARRANGE THE BLOCKS TO SEE….**=

=//There is a space here to link repeated addition to multiplication, and if it is appropriate, I might. (Grade 3 and up) However we OVERUSE repeated addition with students and that creates big issues in the higher grades. If you see 2 + 2 + 2 + 2 and hear someone say thats 2 x 4 I would be inclined to unsnap the cubes and re arrange them to show the areas that are created by 2 x four and 3 x 3. It is too simple to allow young children to think multiplication for addition and becomes unproductive in Grade 4. So any opportunity I have to reinforce that multiplication is an area model and will be seen in a whole different way really matters to me as a teacher.//=



=//Grade ones just parroting back that repeated addition is multiplication is not a stereotype I want to reinforce. I would not introduce multiplication into this discussion with Grade ones or Grade twos. It is not the focus of the work. We are focussed on 2 and 3 make 5 and 5 and 5 make ten.//=